The third and final US presidential debate last week was supposed to focus on foreign policy. Many voters, including myself, plus a global audience expected the debate to offer us a sense of each candidate's strategic vision of the future of American foreign policy.

The debate turned out to be mostly about the US policy towards the Middle East, with a sprinkling of China-bashing, defence spending, and an occasional veering off towards domestic issues. For those of us who think of foreign policy as a set of instruments - diplomacy, military power and development assistance - for international affairs, relations, and the shaping of the world order to come, we expected more from the next leader of the United States.

We expected contrasting visions that were more than tactical approaches to the immediate situations in Syria and the region - though those are important too. In short, we wanted a debate over grand strategy - the fundamental basics of matching ends to means to promote and support the security and interests of the state. There was very little of that.

Romney nonetheless benefited more from the debate than the president. It gave him the opportunity to show the nearly 60 million Americans watching - and hundreds of millions more around the world - his rebranded vision and competence on foreign policy issues. The rebranding now includes a wholesale change in approach towards the Middle East, from a war-ready neo-con hawk to a more pragmatic "realist" - a man of peace and diplomacy. Coming from a Republican candidate, this dovish stance is nothing short of startling.

Before the debate most observers thought Romney was the more ready to go to war to stop Iran gaining a nuclear capability. He was less patient over the use of nonmilitary means such as diplomacy and economic sanctions. The "red line" he had insisted Obama declare has now been replaced by his agreement with Obama's multilateral economic sanctions, which are beginning to bite. Likewise, on Syria, Romney fell in line with Obama by toning down his vow to provide military assistance to the rebels, in favour of working with Russia and China for a political solution. He has also become more kind and generous with pledges of foreign aid to Egypt. This contrasts sharply with the aims of his own running mate, Paul Ryan, who plans to cut spending on foreign aid and foreign affairs by 44 per cent by 2016.

The basic question for voters and the rest of the world is how much of Romney's proclaimed change in foreign policy is trustworthy and believable.

Polls in the US following the debate suggested that Obama won the debate by a small margin due to a more knowledgeable and forceful articulation of his positions on various issues, with Romney seen as ineffective in his attacks. There was little substantive difference, though. The softening of Romney's position on military intervention is more in tune with public sentiment, and that may have helped keep Obama's margin of victory small.

An international poll conducted by the BCC following the debate contrasted sharply with the US polls. The BBC poll of nearly 22,000 people in 21 countries indicated clearly that the rest of the world favours Obama's foreign policy approach. An overall majority of 50 per cent of international respondents supported Obama, compared to only 9 per cent for Romney, with a range as high as 50 to 70 per cent for Obama in countries like Canada, France, Germany, the UK, Brazil, Panama, Indonesia, South Korea, Kenya, and Nigeria. The only exception was Pakistan, where 15 per cent backed Romney and only 11 per cent Obama.

Why does the rest of the world seem not to believe in Romney's transformation? There are three plausible explanations.

The first is Romney's avowed faith in unilateralism over multilateralism when it comes to foreign policy. Romney believes in leading from the front, even if that means going it alone in the mode of the Bush-Cheney approach to Iraq. Obama believes in multilateralism and international legitimacy (or "leading from behind", as Romney and his neo-con hawks like to say).

Second, Romney's insistence on military spending does not seem well thought through, betraying an old Cold War mindset.

Third, significant shaping of foreign policy is done by the advisors around the president. Romney has yet to settle the differences between the neo-cons and the pragmatic "realists" that have been advising him, which has resulted in flip-flopping and incoherent policy stances that do not inspire trust or credibility.

The rest of the world will have to wait till November 6 to see whether Americans prefer a candidate who has been broadly consistent and stood his ground or one who plays along with the tune of the day.

Dr Kiertisak Toh is a member of the economic faculty at Radford University, Virginia and a senior fellow at the Centre for International Development, Sanford School of Public Policy, Duke University, North Carolina.